Mathematical optimization or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.
In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints are not linear equalities or the objective function is not a linear function. An optimization problem is one of calculation of the extrema of an objective function over a set of unknown real variables and conditional to the satisfaction of a system of equalities and inequalities, collectively termed constraints. It is the sub-field of mathematical optimization that deals with problems that are not linear.
AMPL is an algebraic modeling language to describe and solve high-complexity problems for large-scale mathematical computing . It was developed by Robert Fourer, David Gay, and Brian Kernighan at Bell Laboratories. AMPL supports dozens of solvers, both open source and commercial software, including CBC, CPLEX, FortMP, MOSEK, MINOS, IPOPT, SNOPT, KNITRO, and LGO. Problems are passed to solvers as nl files. AMPL is used by more than 100 corporate clients, and by government agencies and academic institutions.
The general algebraic modeling system (GAMS) is a high-level modeling system for mathematical optimization. GAMS is designed for modeling and solving linear, nonlinear, and mixed-integer optimization problems. The system is tailored for complex, large-scale modeling applications and allows the user to build large maintainable models that can be adapted to new situations. The system is available for use on various computer platforms. Models are portable from one platform to another.
SNOPT, for Sparse Nonlinear OPTimizer, is a software package for solving large-scale nonlinear optimization problems written by Philip Gill, Walter Murray and Michael Saunders. SNOPT is mainly written in Fortran, but interfaces to C, C++, Python and MATLAB are available.
The TOMLAB Optimization Environment is a modeling platform for solving applied optimization problems in MATLAB.
The Zuse Institute Berlin is a research institute for applied mathematics and computer science on the campus of Freie Universität Berlin in Dahlem, Berlin, Germany.
MOSEK is a software package for the solution of linear, mixed-integer linear, quadratic, mixed-integer quadratic, quadratically constrained, conic and convex nonlinear mathematical optimization problems. The applicability of the solver varies widely and is commonly used for solving problems in areas such as engineering, finance and computer science.
AIMMS is a prescriptive analytics software company with offices in the Netherlands, United States and Singapore.
Optimization Toolbox is an optimization software package developed by MathWorks. It is an add-on product to MATLAB, and provides a library of solvers that can be used from the MATLAB environment. The toolbox was first released for MATLAB in 1990.
LINDO is a software package for linear programming, integer programming, nonlinear programming, stochastic programming and global optimization.
optiSLang is a software platform for CAE-based sensitivity analysis, multi-disciplinary optimization (MDO) and robustness evaluation. It was originally developed by Dynardo GmbH and provides a framework for numerical Robust Design Optimization (RDO) and stochastic analysis by identifying variables which contribute most to a predefined optimization goal. This includes also the evaluation of robustness, i.e. the sensitivity towards scatter of design variables or random fluctuations of parameters. In 2019, Dynardo GmbH was acquired by Ansys.
Deterministic global optimization is a branch of mathematical optimization which focuses on finding the global solutions of an optimization problem whilst providing theoretical guarantees that the reported solution is indeed the global one, within some predefined tolerance. The term "deterministic global optimization" typically refers to complete or rigorous optimization methods. Rigorous methods converge to the global optimum in finite time. Deterministic global optimization methods are typically used when locating the global solution is a necessity, when it is extremely difficult to find a feasible solution, or simply when the user desires to locate the best possible solution to a problem.
Artelys Knitro is a commercial software package for solving large scale nonlinear mathematical optimization problems.
FEATool Multiphysics is a physics, finite element analysis (FEA), and partial differential equation (PDE) simulation toolbox. FEATool Multiphysics features the ability to model fully coupled heat transfer, fluid dynamics, chemical engineering, structural mechanics, fluid-structure interaction (FSI), electromagnetics, as well as user-defined and custom PDE problems in 1D, 2D (axisymmetry), or 3D, all within a graphical user interface (GUI) or optionally as script files. FEATool has been employed and used in academic research, teaching, and industrial engineering simulation contexts.
ANTIGONE, is a deterministic global optimization solver for general Mixed-Integer Nonlinear Programs (MINLP).
Octeract Engine is a proprietary massively parallel deterministic global optimization solver for general Mixed-Integer Nonlinear Programs (MINLP).
